On The Modal Logics of Some Set-Theoretic Constructions
Tanmay C. Inamdar
Abstract:
In set theory, there are various transformations between models. In
particular, forcing, inner models, and ultrapowers occupy a
fundamental place in modern set theory. Each of these play a different
role. For example, forcing and inner models are typically used to
establish the consistency of statements and the consistency strength
of statements, and ultrapowers are typically used to define various
large cardinal notions, which play the role of a barometer for
consistency strength of statements.
Each of these techniques however, can be seen as a process for
starting with one model of set theory, and obtaining another. Indeed,
it is this aspect of these techniques that we are interested in in
this thesis. Each such method of transforming models of set theory
lends itself to analysis by the techniques of modal logic [Ham03,
HL08], which is the general study of the logic of processes. It is a
recent trend in set theory that research has focussed on these modal
aspects of models of set theory. This is partly due to philosophical
concerns, such as Hamkins’s multiverse view [Ham09, Ham11], Woodin’s
conditional platonism [Woo04], Friedman’s inner model hypothesis
[Fri06], but also due to mathematical concerns, such as to account for
the curious fact that, in some sense, these techniques that we have
mentioned are essentially the only known techniques that set theorists
have to prove independence results.
Concretely, if we fix a particular technique of model-transformation,
we may reasonably ask of a given model of set theory questions of the
following nature: “which statements are always true in all models that
we shall construct by using this technique?”; “which statements can we
always change the truth value of in any model that we shall construct
by using this technique?” etc. Questions of the first sort are the
topic of study of the area of set theory which is known as
absoluteness, whereas questions of the second sort are the topic of
study of the area of set theory known as resurrection. However, in
both these cases, the questions we are asking talk about specific
sentences in the language of set theory. That is, while the answers to
these questions change depending on the type of model-transformation
technique that we are considering, they are not purely questions about
these techniques.
In this thesis, we are (for the most part) not interested in this
interplay between a model-transformation technique and sentences in
the language of set theory, but instead, in the purely modal side of
these techniques. That is, we are interested in understanding the
general principles that are true of these techniques when they are
seen as processes. As an example of the kind of questions that we
shall concern ourselves with, consider: “If φ is a statement that is
true in some model that we construct by using this technique, and ψ is
another statement that is true in some model that we construct by
using this technique, then is it the case that we can construct a
model where both φ and ψ are true by using this technique?”, or “If φ
is true of all models that we shall construct by using this technique,
is φ already true?”. Note that the answers to these questions do not
depend on what φ and ψ are, but only on the nature of these
model-transformation techniques.
These questions were first considered by Hamkins in [Ham03]. In
particular, Hamkins showed that by interpreting the modal operator by
“in all forcing extensions” and the ♦ operator by “in some forcing
extension” one could interpret modal logic in set theory in a very
natural way, and using this interpretation, study the technique of
forcing through the modal lens. Hamkins used this interpretation to
express certain forcing axioms known as maximality principles. These
axioms were meant to capture the essence of models where a lot of
forcing had already occurred, or to quote Hamkins, “anything forceable
and not subsequently unforceable is true”, and relativisations of
‘forceable’ to specific types of forcing notions. It is easily seen
that modal logic provides an elegant way of expressing these
statements using the scheme ♦ φ φ. Hamkins also gave a lower bound of
S4.2 for the modal logic that arises from forcing, the modal logic of
forcing, in this paper. Hamkins’s work on maximality principles has
had many follow ups, the earliest ones being [Lei04] and [HW05].
The first paper devoted entirely to the modal logic of forcing was
[HL08]. In particular, they were able to show that the modal logic of
forcing is S4.2. They also studied various generalisations of the
modal logic of forcing, such as the modal logic of forcing with
parameters, and developed some techniques which modularise the process
of calculating the modal logics of set-theoretic constructions.
In addition to this, in [HL08], various relativisations of modal logic
of forcing were also considered. For example, if we fix a definable
class of partial orders P, and a definition for it, we may interpret
the operator as “in all forcing extensions obtained by forcing with a
partial order in P” and the ♦ operator as “in some forcing extensions
obtained by forcing with a partial order in P” and ask what the modal
logic so obtained, denoted by MLP , is. This line of investigation is
the main topic of study of [HLL], where for many natural classes P,
upper and lower bounds are given for their modal logic. We continue
this line of enquiry in this thesis. In particular, we take P to be
the class of ccc-partial orders, and we study their corresponding
modal logic, MLccc. We are able to improve the upper bound for MLccc
which was obtained in [HL08]. In order to do this, we generalise the
method found there from the case of a single ω1-tree to the case of an
arbitrary finite number of ω1-trees. Along the way, we obtain a
characterisation of Aronszajn trees to which a branch can be added by
ccc forcing which is interesting in its own right, and which also
raises some questions of independent interest.
Another different direction that we pursue is that of looking at a
different technique for relating models, namely that of taking
definable-with-parameters inner models. The germs of this endeavour
can be found in [HL13], where the modal logic of the relation of being
a forcing ground 1 is studied. We are able to compute the exact modal
logic of this relation, though this modal theory was not one which had
been considered in this area before. We obtain this theory by adding
an extra axiom to the well-studied modal theory S4.2 which captures
the property of L, Godel’s constructible universe, being in a sense
the minimal model of ZFC. Our proofs strongly rely on the results from
[HL13].